The approximation numbers of Hardy--type operators on trees

The Hardy operator $T_a$ on a tree $\G$ is defined by $$(T_af)(x):=v(x) \int^x_a u(t)f(t) dt \qquad {for} a, x\in \G.$$ Properties of $T_a$ as a map from $L^p(\G)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy $$(*) \lim_{n\to \infty} na_n(T_a) = \alpha_p\int_{\G} |uv|dt$$ for a specified constant $\alpha_p$ and $1<p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, (*) was unknown even when $\G$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.